Dear Dirichlet,

My flatmates and I put up our square-shaped artificial Christmas tree last week and decorated it beautifully. However, when I get home from work, I find it on its side on the floor. I think my flatmates are pushing it over but it doesn’t matter how much I shout at them, they insist it’s not their fault. Do you know why this is happening?

— Tinselitus, Glasgow

Dirichlet says:

Check the determinant of your quadratic tree. You should find that the solutions to your tree are of the form

$$holly + \mathrm{i} v,$$

where $\mathrm{i} = \sqrt{-1}$ and $holly, v \in \mathbb{R}$. You are probably having problems because $v \neq 0$, i.e. your tree has no real roots.

Dear Dirichlet,

I’m cooking our annual Christmas dinner for my friends this year and I’m worried that it’s going to be a disaster. A few years ago there were only six of us, which I could handle, but since then our friends have starting bringing their friends, and this year they’re all bringing plus-ones. I’m terrified that the numbers are multiplying out of control. Am I stuffed?

— R. Cratchet, London

Dirichlet says:

Don’t worry, Bob. Multiplying numbers of guests is a scary proposition: maybe you would be happier with just a small addition of guests instead? Try logarithms to turn multiplication problems into simple addition. In fact, I highly recommend for this time of year the yule log.

Dear Dirichlet,

My dog looked a little under the weather a week ago and a friend of mine suggested that I should find something to perk him up. I was looking for something natural, and the herbal remedies shop on the high street sold me this clearish liquid which I was supposed to add a few drops of to his water bowl. Anyway, yesterday I woke up and found that he has two bumps on the top of his head. And by this morning, they’d grown to a couple of inches in height! I think he’s growing antlers?!

— Ted Maltin, Twin Cities, Minnesota

Dirichlet says:
This is fixable but you need to act now. First, stop using modular arithmetic: remove all the clocks and pianos from your house. Then get rid of any congruences in your dog’s life: make sure all his biscuits are different shapes and toss away any copies of Despicable Me. Finally, take a few days off from work: if you stop commuting, the effect should break down.

It should clear up in a few days but you should have been more careful. Looks like you bought some dodgy Chinese Reindeer Serum.

Dear Dirichlet,

My wife tells me that my gift presentation skills often leave a bit to be desired. You seem like a man who has Christmas all wrapped up; do you have any Christmas present packaging tips?

— Chris Moss, Troy

Dirichlet says:

Let $X$ be the present. First, prove the existence of a compact covering space over $X$ embedded in a suitable subspace of $\mathbb{R}^n$. The rest is trivial and left as an exercise for the reader.